Looking at the Taylor Series for the following functions around 0.

1.) \(f(x)=\frac{1}{1-x}\)

\[ \text{Let } u=1-x\\ \frac{d}{dx}f(x)=\frac{d}{du}\frac{1}{u}\frac{du}{dx}=\\ -\frac{1}{u^2}\cdot-1=\frac{1}{u^2}=\frac{1}{(1-x)^2}\\ \frac{d^2}{dx^2}f(x)=\frac{2}{(1-x)^3}\\ \frac{d^3}{dx^3}f(x)=\frac{6}{(1-x)^4}\\ \frac{d^n}{dx^n}f(x)=\frac{n!}{(1-x)^{n+1}} \]

Putting this in around 0, we get:

\[ f(x)=f(0)+f^\prime(0)x+f^{\prime\prime}(0)\frac{x^2}{2!}+f^{\prime\prime\prime}(0)\frac{x^3}{3!}+\dots=\\ 1+x+\frac{2!}{1}\cdot\frac{x^2}{2!}+\frac{3!}{1}\cdot\frac{x^3}{3!}+\frac{4!}{1}\cdot\frac{x^4}{4!}+\dots=\\ \sum_{n=0}^\infty x^n \]

This one is interesting, as I learned this in the opposite direction, as the solution to \(\sum_{n=0}^{\infty} x^n\) for\(|x|<1\)

f1<-function(x) 1/(1-x)
t<-taylor(f1,0,4)
t
## [1] 1.000029 1.000003 1.000000 1.000000 1.000000

2.) \(f(x)=e^x\)

\[ \frac{d}{dx}f(x)=f(x) \]

Putting this around 0

\[ f(0)=1 \]

\[ f(x)=f(0)+f^\prime(0)x+f^{\prime\prime}(0)\frac{x^2}{2!}+f^{\prime\prime\prime}(0)\frac{x^3}{3!}+\dots=\\ 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots=\\ \sum_{n=0}^\infty\frac{x^n}{n!} \]

f2<-function(x) exp(x)
taylor(f2,0,8)
## [1] 2.505533e-05 1.961045e-04 1.386346e-03 8.334245e-03 4.166657e-02
## [6] 1.666667e-01 5.000000e-01 1.000000e+00 1.000000e+00
sum(taylor(f2,0,8))
## [1] 2.718275

This is very close to the value of \(e\).

3.) \(f(x)=\ln(x+1)\)

\[ \frac{d}{dx}f(x)=\frac{1}{x+1}\\ \frac{d^2}{dx^2}f(x)=-\frac{1}{(x+1)^2}\\ \frac{d^3}{dx^3}f(x)=\frac{2}{(x+1)^3}\\ \frac{d^4}{dx^4}f(x)=-\frac{6}{(x+1)^4} \]

In general

\[ \frac{d^n}{dx^n}f(x)=\frac{(n-1)!}{(x+1)^n} \]

Around 0

\[ f(0)=0\\ f^\prime(0)=1\\ f^{\prime\prime}(0)=-1 \]

So we have

\[ f(x)=f(0)+f^\prime(0)x+f^{\prime\prime}(0)\frac{x^2}{2!}+f^{\prime\prime\prime}(0)\frac{x^3}{3!}+\dots=\\ 0+x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots=\\ \sum_{n=1}^\infty(-1)^{n+1}\frac{x^n}{n} \]

f3<-function(x) log(x+1)
taylor(f3,0,8)
## [1] -0.1272487  0.1436357 -0.1668792  0.2000413 -0.2500044  0.3333339
## [7] -0.5000000  1.0000000  0.0000000